# Imperfect not implies hypoabelian

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., imperfect group) neednotsatisfy the second group property (i.e., hypoabelian group)

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This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., imperfect group) neednotsatisfy the second group property (i.e., solvable group)

View a complete list of group property non-implications | View a complete list of group property implications

Get more facts about imperfect group|Get more facts about solvable group

## Statement

It is possible for a group to be an imperfect group but not a hypoabelian group. In particular, it is possible for an imperfect group to not be a solvable group.

## Proof

### Finite example

For a finite group, being hypoabelian is equivalent to being a finite solvable group, so it suffices to construct a finite imperfect group that is not solvable.

The smallest example is symmetric group:S5:

- The group is imperfect: As we can see from the subgroup structure of symmetric group:S5, the only normal subgroups are the trivial group, A5 in S5, and the whole group. The only nontrivial quotients are thus the whole group and cyclic group:Z2. The group itself is not perfect because its derived subgroup is A5 in S5. The quotient cyclic group:Z2 is also not perfect. Thus, the group is imperfect.
- The group is not solvable, and hence not hypoabelian: This can be seen in many ways. For instance, the derived series stabilizes at A5 in S5, and does not reach the trivial group. Alternatively, we note that the group has a simple non-abelian composition factor, namely alternating group:A5, hence is not solvable.